published_or_final_version / Mathematics / Master / Master of Philosophy

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In this thesis, we show that several natural questions about automatic sequences can be expressed as logical predicates and then decided mechanically. We extend known results in this area to broader classes of sequences (e.g., paperfolding words), introduce new operations that extend the space of possible queries, and show how to process the results. We begin with the fundamental concepts and problems related to automatic sequences, and the corresponding numeration systems. Building on that foundation, we discuss the general logical framework that formalizes the questions we can mechanically answer. We start with a first-order logical theory, and then extend it with additional predicates and operations. Then we explain a slightly different technique that works on a monadic second- order theory, but show that it is ultimately subsumed by an extension of the first-order theory. Next, we give two applications: critical exponent and paperfolding words. In the critical exponent example, we mechanically construct an automaton that describes a set of rational numbers related to a given automatic sequence. Then we give a polynomial-time algorithm to compute the supremum of this rational set, allowing us to compute the critical exponent and many similar quantities. In the paperfolding example, we extend our mechanical procedure to the paperfolding words, an uncountably infinite collection of infinite words. In the following chapter, we address abelian and additive problems on automatic sequences. We give an example of a natural predicate which is provably inexpressible in our first-order theory, and discuss alternate methods for solving abelian and additive problems on automatic sequences. We close with a chapter of open problems, drawn from the earlier chapters.

For young infants, knowledge of physical objects and animate agents seems highly rigid, with no information combined across domains. Adult cognition, however, is more flexible. In this thesis, I use a special category of object—a tool—that can only be reasoned about appropriately if information is combined across domains. Using this special case, I examine whether older infants are capable of integrating functional information about the tool while making inferences about the intent of the tool-user. Experiment 1 shows that infants can reason about complex means-end sequences involving tools; and Experiments 2 and 3 both show that under some circumstances, infants can take into account functional information about the tool when making these sorts of inferences. Together, these studies extend previous findings about how infants understand complex means-end sequences, and demonstrate that by 13 months, infants are already combining knowledge across domains. / Thesis (Master, Psychology) -- Queen's University, 2007-08-28 11:26:07.974

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The Erdős discrepancy problem asks, "Does there exist a sequence t = {t_i}_{1≤i<∞} with each t_i ∈ {-1,1} and a constant c such that |∑_{1≤i≤n} t_{id}| ≤ c for all n,c ∈ ℕ = {1,2,3,...}?" The discrepancy of t equals sup_{n≥1} |∑_{1≤i≤n} t_{id}|. Erdős conjectured in 1957 that no such sequence exists. We examine versions of this problem with fixed values for c and where the values of d are restricted to particular subsets of ℕ. By examining a wide variety of different subsets, we hope to learn more about the original problem. When the values of d are restricted to the set {1,2,4,8,...}, we show that there are exactly two infinite {-1,1} sequences with discrepancy bounded by 1 and an uncountable number of in nite {-1,1} sequences with discrepancy bounded by 2. We also show that the number of {-1,1} sequences of length n with discrepancy bounded by 1 is 2^{s2(n)} where s2(n) is the number of 1s in the binary representation of n. When the values of d are restricted to the set {1,b,b^2,b^3,...} for b > 2, we show there are an uncountable number of infinite sequences with discrepancy bounded by 1. We also give a recurrence for the number of sequences of length n with discrepancy bounded by 1. When the values of d are restricted to the set {1,3,5,7,..} we conjecture that there are exactly 4 in finite sequences with discrepancy bounded by 1 and give some experimental evidence for this conjecture. We give descriptions of the lexicographically least sequences with D-discrepancy c for certain values of D and c as fixed points of morphisms followed by codings. These descriptions demonstrate that these automatic sequences. We introduce the notion of discrepancy-1 maximality and prove that {1,2,4,8,...} and {1,3,5,7,...} are discrepancy-1 maximal while {1,b,b^2,...} is not for b > 2. We conclude with some open questions and directions for future work.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2009. / Title from title screen (site viewed February 25, 2010). PDF text: ii, 93 p. : ill. ; 1 Mb. UMI publication number: AAT 3386580. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph.D.)--Washington State University. / Includes bibliographical references.

Thesis (M. Phil.)--University of Hong Kong, 1994. / Includes bibliographical references (leaves 35-36).